I am curious if there is any literature (texbooks, mainly, but articles will do too, though I don't have easy access to any paid journal) that deals with general relativity by using Ehresmann connections on the (orthonormal) frame bundle in a rigorous manner, rather than Koszul connections on the tangent bundle, and develops calculus directly on the frame bundle, rather than on spacetime itself.
Basically, I am interested in this, for the sake of being interested in it, however I have hopes that I might be able to use this formalism to attack some problems in my research.
To clarify a bit more:
- I am not looking for local tetrad formalism, however I am hoping that if what I am asking for exists, it will resemble local tetrad formalism a lot, but with globally defined quantities, as opposed to just local ones.
- Although I'll take any materials on the subject gladly, I really would prefer if the resource treated lagrangian formalism. I am mostly aware of how Ehresmann connections on the frame bundle work, but I have absolutely no clue how to do langrangian formalism with it, and this is an absolute necessity for my work.
- I have Kobayashi & Nomizu for necessary extra mathematical details, but I'd prefer this resource to be generally self-contained.
Best Answer
A first remark is that in many spacetimes of interest, it is possible to choose a global tetrad (or frame field). So the need to lift everything from the spacetime to the frame bundle to have globally defined objects disappears. This is the case, for example, on any globally hyperbolic spacetime where the Cauchy surface is a prallelizable manifold. All compact orientable 3-manifolds and even many non-compact ones are parallelizable.
The above observation might explain why most references don't bother going beyond the local tetrad formalism. However, I do know of at least two references that bother going through the exercise of lifting all the relevant objects to the frame bundle:
Both papers are rather extensive and only some of the early sections might be relevant for what you are interested in.
While both these references are quite recent, I'm sure that the method of working directly on the frame bundle has been known for a long time. I don't know though who might have been the first to go through a similar exercise in the literature.